On page 342, Ricklefs tells how Pearl and Reed in 1920 used the logistic model of population growth to predict future densities of the United States population.

1. What is the current estimated size of the US population, as given by the US Census Bureau? On what day and time did you obtain this estimate?

1790	1800	1810	1820	1830	1840	1850
3,929,214	5,308,483	7,239,881	9,638,453	12,866,020	17,069,453	23,191,876
1860	1870	1880	1890	1900	1910	1920
31,443,321	38,558,371	50,189,209	62,979,766	76,212,168	92,228,496	106,021,537
1930	1940	1950	1960	1970	1980	1990
123,202,624	132,164,569	151,325,798	179,323,175	203,302,031	326,542,199	248,709,873

Use the same values of r (0.03134 persons per parent per year) and K (197,273,000 persons) used by Pearl and Reed in the logistic model to predict census counts for these years. Enter your predicted values in the text box below. If you used a spreadsheet to predict these values, copy and paste the spreadsheet table into the textbox.

Suggestion: An easy way to calculate the predicted values is to use a spreadsheet on your computer, e.g., Excel, QuattroPro, etc. Put the value of r in one cell and the value of K in the other. Use a third cell to hold the value of b. Put years in one column (starting with 0 for the 1790 census, then 10 for 1800, etc.) and the census counts from the above table in the next column.

In the first cell of the next column put a formula for the logistic prediction for 1790. When you enter the expression for the logistic, use the cell references for the cells containing r, K, and b, not the values themselves. Both Excel and QuattroPro have the EXP and LN functions. EXP raises the constant, e, to a power (EXP(r) gives you e^r ). LN takes the natural log of a number (LN(lambda) gives you log_e(lambda)). Don't forget to put in the minus sign in the EXP function for your predicted values from the logistic equation.

After you calculate the predicted value for 1790, edit the formula so that you make the references to the cells containing r, K, and b absolute (put a $ in front of the column and the row ID's). Click on help if you need more information on absolute references. Then copy the formula down the column to fill in the other 20 cells. (In Excel, click on the first cell in the column containing the formula. Hold down the left mouse button and drag the cursor down the column as far as you want the formula to go (i.e., to 1990). Release the mouse button and Excel will duplicate the formula in all those cells. That will save you lots of time.)

If you don't have a spreadsheet on your computer, you can use a calculator. If you use a calculator, store the values of r, K, and b in different registers if you can. Don't round off until the last step in your calculations.

Suggestion: To make a chart in Excel using the Chart Wizard, first use the mouse to select the cells you want to graph (put the cursor on the first cell you want (i.e., the census count for 1790), click the left mouse button, and drag across columns and rows to the last cell in the group (i.e., the predicted census for 1990)).

Click on the chart icon on the tool bar, or Click on Insert, Chart. Pick the Line chart from the choices. Click next and you will see what the graph will look like. Click the Series tab. Click in the Name window and enter the name of the series, e.g., Census. In the Category (x) axis labels, click on the table icon at the right of the window. Put the cursor on the cell with 0 years and drag it down the column. Let the mouse button up and type the Enter button. Notice that the years are now on the x-axis. Click on Series 2 and enter its name, e.g., Logistic.

Click on next and enter appropriate titles for your graph and the axes. Select to Place chart As new sheet and enter a name if you want. Right click in the gray area of the chart and pick Format Plot Area. You can make it white, which saves toner in the printer.

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